Efficient knot group identification as a tool for studying entanglements of polymers

Abstract

A technique is presented for the identification of the knot group of knots, links, and other embedded graphs as a tool in numerical studies of entanglements of polymers. With this technique, the knot group is simultaneously more discriminating and easier to calculate than the knot invariants that have been used in such studies in the past. It can be applied even in cases of very complex knot projections with hundreds of crossings. Starting from an arbitrary projection of an embedded graph, we generate a sequence of representations, any one of which is a full and complete representation of the knot group. Any two knot groups are isomorphic if they have identical representations. Therefore, we compare the sequence of representations of any given knot or link against a previously determined lookup table, and if the group of the knot or link is represented in this table we eventually find a match and identify the knot group.